## 5.7 Other controlled gates

### 5.7.1 Controlled-phase

Needless to say, not everything is about the controlled-**controlled-phase** gate

controlled-phase |
---|

We can also represent the

Again, the matrix is written in the computational basis **controlled- Z gate**, which acts as

In order to see the entangling power of the controlled-phase shift gate, consider the following circuit.

(Generating entanglement, again).

In this circuit, first the two Hadamard gates prepare the equally-weighted superposition of all states from the computational basis

and then the controlled-

which results in the entangled state.

### 5.7.2 Controlled-U

Both the above two-qubit controlled gates (i.e. **controlled- U gate**:

controlled- |
---|

We can also represent the

We can go even further and consider a more general unitary operation: the two-qubit ** x-controlled-U gate**:

### 5.7.3 Phase kick-back

Before moving on to the next section, we first describe a simple “trick” — an unusual way of introducing phase shifts that will be essential for our analysis of quantum algorithms. Consider the following circuit.

(Controlled-

where *eigenstate* of

This should look familiar: it is the usual interference circuit, but with the phase gate replaced by a controlled-*required to be an eigenstate of U*.
The circuit effects the following sequence of transformations (omitting the normalisation factors):

*not*get entangled with the first one: it remains in its original state

Consider the following

Now, prepare the qubit in the second register in state *the phase kick-back mechanism introduced a relative phase in the equally-weighted superposition of all binary strings of length two*.

Phase kick-back is how we control quantum interference in quantum computation.

We will return to this topic later on, when we discuss quantum evaluation of Boolean functions and quantum algorithms.

### 5.7.4 Universality, revisited

We will come across few more gates in this course, but at this stage you already know all the elementary unitary operations that are needed to construct any unitary operation on any number of qubits:

- the Hadamard gate,
- all phase gates, and
- the
\texttt{c-NOT}

These gates form a **universal set of gates**: with ^{81}
We should mention that there are many universal sets of gates.
In fact, almost any gate that can entangle two qubits can be used as a universal gate.

We are particularly interested in any *finite* universal set of gates (such as the one containing the Hadamard,

### 5.7.5 Density operators and the like

The existence of entangled states leads to an obvious question: if we cannot attribute a state vector to an individual qubit, then how can we describe its quantum state? In the next few chapters we will see that, when we limit our attentions to a part of a larger system, states are not represented by vectors, measurements are not described by orthogonal projections, and evolution is not unitary. As a spoiler, here is a dictionary of some of the new concepts that will soon be introduced:

state vectors | density operators | |

unitary evolutions | completely-positive trace-preserving maps | |

orthogonal projectors | positive operator-valued measures |

Recall the big-

O asymptotic notation: given a*positive*functionf(n) , we writeO(f(n)) to mean “bounded above byc\,f(n) for some constantc > 0 (for sufficiently largen )”. For example,15n^2+4n+7 isO(n^2) .↩︎